In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either $+1$ or $−1$ and whose rows are mutually orthogonal.
Questions tagged [hadamard-matrices]
53 questions
7
votes
3 answers
Who conjectured that there are only finitely many biplanes, and why?
This question on MathOverflow motivates me to ask what the reasoning is behind the conjecture that there are only finitely many biplanes. More generally, it has been conjectured that for fixed $\lambda>1$ there are only finitely many triples…
Will Orrick
- 19,218
7
votes
2 answers
Is there any proof that there doesn't exist a circulant Hadamard matrix of size $8 \times 8$?
Is there any proof that there doesn't exist an $8 \times 8$ circulant Hadamard matrix?
A matrix $H \in \{\pm 1\}^{n \times n}$ is Hadamard if $H H^T = n I$, where $I$ is the $n \times n$ identity matrix. Then, a Hadamard matrix $H$ such that…
Danny_Kim
- 3,553
6
votes
1 answer
What is the relationship between two-level full factorial designs and Hadamard matrices?
Background
While studying two-level full factorial designs and Hadamard matrices, I've noticed several commonalities between these mathematical structures. In particular, both involve matrices with elements of $\pm 1$ and share properties related to…
6
votes
1 answer
Another Hadamard matrix of order 4?
Wikipedia states that there is, up to equivalence, a unique Hadamard matrix of order 4, namely
$$
\def\p{\phantom+}
\begin{pmatrix}
\p1&\p1&\p1&\p1 \\
\p1&-1&\p1&-1 \\
\p1&\p1&-1&-1 \\
\p1&-1&-1&\p1
\end{pmatrix}.$$
As equialence operations are…
M. Rumpy
- 1,035
6
votes
1 answer
Is there any proof that there doesn't exist a Hadamard matrix of size $6 \times 6$?
A matrix $H \in {\pm 1}^n $ is Hadamard matrix if $HH^T=nI_n$, where $I$ is $n\times n$ identity matrix. Hadamard's conjecture said that there exists Hadamard matrix of order 1,2 or $4n$, for every positive integer $n$.
My question is how to prove…
Riris
- 61
5
votes
1 answer
Construction of Hadamard Matrices of Order $n!$
I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power.
Since
$$
n!-1 \bmod 4 = 3
$$
construction 1 has to be chosen:
If $q$ is…
draks ...
- 18,755
4
votes
2 answers
Is there a 12x12 symmetric Hadamard matrix?
More generally, is there a simple condition for which $n$ there are symmetric Hadamard matrices of order $n$?
This set of $n$ is closed under multiplication via the Kronecker product.
chfb
- 63
- 3
4
votes
1 answer
Efficient matrix-vector multiplication for "partial" Hadamard matrices
I've recently been working on an algorithm for bilinear systems in the form $y = (Lw) \odot (Rx)$, where $\odot$ denotes the elementwise product between two matrices of compatible sizes. In the above, we assume $w \in \mathbb{R}^d$ and $L \in…
VHarisop
- 4,100
4
votes
1 answer
Hadamard Form of a Circulant Matrix
Definition: Let a field $\mathbb{F}$. Consider an $2^n \times 2^n$ matrix $\bf H$ over $\mathbb{F}$. $\bf H$ is called Hadamard over $\mathbb{F}$ if and only if
$$
{\bf H}=\left(
\begin{array}{cc}
{\bf U} & {\bf V} \\
{\bf V} & {\bf…
Amin235
- 1,905
3
votes
0 answers
Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices
Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2
,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume all diagonal entries are positive: $d_i >0$ and…
Ruben Verresen
- 1,476
3
votes
3 answers
Showing only if for the Hadamard Matrix Equality case with |a| < 1
Let $H = (h_{ij})$ be a square matrix of order $n$ such that $|h_{ij}| \leq 1$. Then, by the Hadamard Determinant inequality we know that $$|\det(H)| \leq n^{\frac{n}{2}}$$
I read in this paper that, equality is achieved if and only if $H$ is a…
Robertmg
- 2,155
3
votes
0 answers
Maximum number of binary n-vectors (i.e., vertices of the n-hypercube) with a given pairwise Hamming distance
Given positive integers $n$ and $d \le n$, what is the maximum possible number $M(n, d)$ of binary $n$-vectors such that the Hamming distance between each pair of those vectors is exactly $d$? For instance, if $n = 3$ and $d = 2$, the answer is $M =…
Luka Fürst
- 31
3
votes
1 answer
Small question about symmetric Hadamard matrices
I am trying to understand Hadamard matrix.
My question is simply about whether Hadamard matrix can always be made equivalent (by multiplying row/column by -1 or swapping rows/columns) to a symmetric one?
I only tried a couple but could not find…
10understanding
- 747
3
votes
2 answers
Proof that Hadamard matrices of order $4k+2$ don't exist
It's known that Hadamard matrices can only exist for orders $1$, $2$ and $4k$. It's easy to show that there are no Hadamard matrices of order $2k+1$. But what is the proof that there are no Hadamard matrices of order $4k+2$?
emptysamurai
- 328
3
votes
0 answers
Is There a Unitary Transformation Which Maps Non-Orthogonal Vectors into 'Uniform' Vectors?
I would like to find a unitary matrix $U$ which maps normalised vectors $\mathbf{v}^{(i)}$ from the set:
\begin{equation}
\mathcal{V} = \Bigg\{
\begin{pmatrix}
1 \\
0\\
0
\\
\vdots
\end{pmatrix},
\frac{1}{\sqrt{2}}
\begin{pmatrix}
1…
Patrick
- 31
- 3